As the temperature is raised from 20°C to 40°C, the average kinetic energy of neon atoms changes by

Q: Why can't I just take the ratio of $40^\circ\text{C}$ and $20^\circ\text{C}$ to get 2?

The formula for kinetic energy is derived using the absolute temperature scale (Kelvin). Zero kinetic energy corresponds to $0\text{ K}$, not $0^\circ\text{C}$. Using Celsius will give you completely incorrect ratios.

Q: Does the type of gas (Neon) matter for average kinetic energy?

No! The average translational kinetic energy of any ideal gas molecule is $\frac{3}{2} k_B T$, regardless of its mass or whether it's a noble gas like Neon. It depends *only* on temperature.

Q: What is the difference between average kinetic energy and total kinetic energy?

Average kinetic energy refers to the energy of a single molecule ($\frac{3}{2} k_B T$). Total kinetic energy refers to the energy of the entire gas sample ($\frac{3}{2} n R T$), which also depends on the number of moles.

Q: Why is option (b) $\sqrt{313/293}$ incorrect?

Option (b) would be the correct answer if the question asked for the change in *root mean square velocity* ($v_{\text{rms}}$), because $v_{\text{rms}} \propto \sqrt{T}$. But for kinetic energy, it's directly proportional to $T$.

Q: How do I quickly convert Celsius to Kelvin in exams?

Simply add $273$ to the Celsius value. While the exact value is $273.15$, using $273$ is standard practice in JEE/NEET unless specified otherwise.

Question

As the temperature is raised from $20^\circ	ext{C}$ to $40^\circ	ext{C}$, the average kinetic energy of neon atoms changes by a factor of which of the following?

Options:

(A) $1/2$

(B) $\sqrt{313/293}$

(C) $313/293$

(D) $2$

Sigma Percentile · Accepted Answer

### The Heat is On: Understanding Kinetic Energy in Gases

Imagine a closed container filled with neon gas. Right now, it is sitting at a comfortable $20^\circ	ext{C}$. The atoms are zipping around, bouncing off the walls in a state of constant, random motion. Now, we turn up the heat, raising the temperature to $40^\circ	ext{C}$. What exactly happens to the energy of these atoms? Let's dive into the kinetic theory of gases to find out.

### The Master Equation

To understand the change in energy, we need our master equation from the kinetic theory of gases. The average translational kinetic energy of a single gas atom is given by:

$$E_k = \frac{3}{2} k_B T$$

where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature. Notice how the energy depends *only* on the temperature. It doesn't matter that the gas is neon; the formula applies universally to any ideal gas.

### The Absolute Temperature Trap

Here is where a classic mistake happens. A lot of students will just take the ratio of $40$ and $20$ and confidently declare that the energy has doubled. But wait! In thermodynamics, temperature must always be measured in **Kelvin**.

If you use Celsius, you will fall right into the trap. Zero kinetic energy corresponds to absolute zero ($0	ext{ K}$), not $0^\circ	ext{C}$. Therefore, we must convert our temperatures:

$$T_1 = 20 + 273 = 293 	ext{ K}$$

$$T_2 = 40 + 273 = 313 	ext{ K}$$

### Final Calculation

Since the kinetic energy is directly proportional to the absolute temperature ($E_k \propto T$), the factor by which it changes is simply the ratio of the final temperature to the initial temperature. Let's substitute our Kelvin values:

$$\frac{E_{k2}}{E_{k1}} = \frac{T_2}{T_1} = \frac{313}{293}$$

And there we have it! The average kinetic energy of the neon atoms increases by a factor of $\frac{313}{293}$. It doesn't double; it just increases by a small fraction. This perfectly matches option (c).

### Food for Thought

Before we wrap up, think about this: What if the examiner asked for the factor by which the **root mean square velocity** ($v_{	ext{rms}}$) changes?

Since $v_{	ext{rms}} = \sqrt{\frac{3RT}{M}}$, velocity is proportional to the square root of temperature ($v_{	ext{rms}} \propto \sqrt{T}$). In that scenario, the answer would have been $\sqrt{\frac{313}{293}}$, which corresponds to option (b). Always read the question carefully to see whether it asks for energy or velocity!

Animated Solution for Chemistry - States of Matter: As the temperature is raised from $2 0^{\circ} C$ to $4 0^{\circ} C$ , the average kinetic energy of neon atoms changes by a factor of which of the following?

Select Answer:

Visualized Solution

$T_{1} \to T_{2}$

$E_{k} = \frac{3}{2} k_{B} T$

$T (K) = T (^{\circ} C) + 273$

$T_{1} = 293 K, T_{2} = 313 K$

\frac{E_{k2}}{E_{k1}} = \frac{T_2}{T_1}

\text{Factor} = \frac{313}{293}

$v_{rms} \propto T$

The Sigma Insight: Gaseous State

The Heat is On

The Master Equation

The Absolute Temperature Trap

Final Calculation

Food for Thought

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